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\title{Homework 0}

\author{Li Zhiqi\quad 3180103041 , }

\begin{document}
\maketitle
\section{Exercise A.15.}
A function $f$:\ $\mathbb{R}\rightarrow\mathbb{R}$ is NOT \textit{Riemann integrable} on $[a,b]$ iff
\begin{eqnarray}
  &\forall L \in \mathbb{R},\exists\ \epsilon_0 > 0, s.t.\  \forall\ \delta > 0,\nonumber\\
  &\exists\ P_N(a,b)\  with\  h(P_N)<\delta, s.t. \nonumber\\
  &|S_N(f)-L|\geq \epsilon_0.
\end{eqnarray}

\section{Exercise A.19.}
(a)\ Set $\mathbb{P} = \{prime\}$, then\\
\quad Origin:\quad $\exists ! p \in \mathbb{P}\cap 2\mathbb{N}^+,\ p = 2.$\\
\quad Negation:\quad $\exists p \in \mathbb{P}\cap 2\mathbb{N}^+,\ p \ne 2.$\\
(b)\ \\
\quad Origin:\quad $\forall a,b,c \in \mathbb{Z}, (ab)c = a(bc).$\\
\quad Negation:\quad $\exists a_1,b_1,c_1 \in \mathbb{Z}, (a_1b_1)c_1 \ne a_1(b_1c_1).$\\
(c)\ Set $A = \{a:a\in2\mathbb{N}^++2,\forall p,q\in\mathbb{P},a\ne p+q\}$,then\\
\quad Origin:\quad $|A| < \infty$.\\
\quad Negation:\quad $|A| = \infty$.

\section{Exercise B.110.}
$\forall\mathbf{u,v,w}\in\mathcal{V}$, we have
\begin{eqnarray}
 & \langle\mathbf{u},\mathbf{v}+\mathbf{w}\rangle=\overline{\langle\mathbf{v}+\mathbf{w},\mathbf{u}\rangle}=\overline{\langle\mathbf{v},\mathbf{u}\rangle}+\overline{\langle\mathbf{w},\mathbf{u}\rangle}\nonumber\\
 &=\langle\mathbf{u},\mathbf{v}\rangle+\langle\mathbf{u},\mathbf{w}\rangle.
\end{eqnarray}
$\forall a \in\mathbb{F},\forall\mathbf{v,w}\in\mathcal{V}$, we have
\begin{eqnarray}
   \langle\mathbf{v},a\mathbf{w}\rangle & = \overline{\langle a\mathbf{w},\mathbf{v}\rangle}=\overline{a\langle\mathbf{w},\mathbf{v}\rangle}\nonumber\\
  &=\overline{a} \ \overline{\langle\mathbf{w},\mathbf{v}\rangle}=\overline{a}\langle\mathbf{v},\mathbf{w}\rangle.
\end{eqnarray}

\section{Exercise C.42.}
If $a>0$,$\forall\epsilon>0,\exists \delta = \frac{a^3\epsilon}{2}$, s.t.\\
$|x-y| < \delta \Rightarrow |f(x)-f(y)|= \frac{|x^2-y^2|}{x^2y^2}\leq|x-y|(|\frac{x}{x^2y^2}|+|\frac{y}{x^2y^2}|)<|x-y|\frac{2}{a^3}<\epsilon.$\\
If $a = 0$, we will prove:\ $\forall\epsilon>0,\forall\delta>0, \exists x,y>0, s.t. |x-y|\leq\delta \Rightarrow |\frac{1}{x^2}-\frac{1}{y^2}|\geq\epsilon$.\\
If $\delta\geq\frac{1}{2\sqrt{\epsilon}}$, we choose $x = \frac{1}{\sqrt{\epsilon}},y = \frac{1}{2\sqrt{\epsilon}} $, then $|x-y|\leq\delta$ and $|f(x)-f(y)| = 3\epsilon > \epsilon$.\\
If $\delta<\frac{1}{2\sqrt{\epsilon}}$, we choose $x \in (0,\sqrt{\epsilon}\delta^2), y \in (2\sqrt{\epsilon}\delta^2,\delta)$, then $|x-y|>\sqrt{\epsilon}\delta^2$ but $|x-y|< \delta$. Hense $|f(x)-f(y)| = \frac{|x^2-y^2|}{x^2y^2}> \frac{|x+y|}{x^2y^2}\sqrt{\epsilon}\delta^2 > \frac{2\sqrt{\epsilon}\delta^2\sqrt{\epsilon}\delta^2}{\epsilon\delta^4\delta^2}=\frac{2}{\delta^2}>8\epsilon>\epsilon$.


\section{A story about
determinants }
In this part we will discuss about determinants. Determinant is a functional mapping a matrix with size $n\times n$ to a real number. It shows the change of length and direction when the matrix(or linear transformation) acts on a vector, and it's widely used in affine transformation analysis, linear equation solving, diagonalization theory and so on.\\
The idea of determinant comes from signed volume. Therefore, the determinant has a geometric meaning: the generalized volume of the parallelotope formed by the vector group of the matrix. We select several properties with geometric significance as the definition of signed volume, namely identity element, sign changing  and linearity. By using these three properties, we can get the signed volume of any parallelotope. In particular, we can determine the sign of the signed volume through a series of permutations aiming at identity element. With the above preparations, we can check that the definition of the determinant(Leibniz formula of determinants) is consistent with the signed volume, which verifies the geometric meaning of the determinant.\\
Finally we list the relationship about several related concepts. Signed volume of parallelotope is a geometric concept,
and determinant is a mathematical abstraction of signed volume. In addition, the cofactor of matrix and minor of matrix
are special determinants.\\
 

\end{document}
